Musical instrument strings with variable thickness.

[Dave]

I was reading this:
http://www.dailymail.co.uk/sciencetech/article-2703107/Now-THATS-string-theory-Physicist-reveals-science-make-better-guitar-player.html
and was wondering about guitar strings with variable thickness along its
length. Could you have interesting harmonics with a well designed string.

Would you get a nice sound or just a dull metallic twang with no sustain?

Plus also, can someone kindly tell me what mu naught stands for in the
equation on the aforementioned webpage.

[Frank Colessi]

U_0 is the mass density of the string, which determines the wave velocity:
v = sqrt(T/u_0).

For a string with variable thickness, u_0 would no longer be a constant
but would be a function of the position, u_0(x), on the string. The
resulting wave equation would be non-linear and depending on the form
of u_0(x) may be impossible to solve explicitly.

But the resulting standing wave pattern on a string of variable thickness
will likely be similar to a sine wave with increasing amplitude and
decreasing frequency along the length of the string (from thin to thick).

My guess is that the sound from such standing waves would be harsh due
to the closeness of the frequencies generated. I am not sure how the
bridge and sounding board would interact with such a standing wave.

Such a string could be easily made by wrapping one string with another
while increasing the winding distance along the length.

[Scott Dorsey]

Frank Colessi <n...@nowhere.net> wrote:
>But the resulting standing wave pattern on a string of variable thickness
>will likely be similar to a sine wave with increasing amplitude and
>decreasing frequency along the length of the string (from thin to thick).
>
>My guess is that the sound from such standing waves would be harsh due
>to the closeness of the frequencies generated. I am not sure how the
>bridge and sounding board would interact with such a standing wave.
>
>Such a string could be easily made by wrapping one string with another
>while increasing the winding distance along the length.

Which is in fact done with some commercial bass strings, and they are in
fact brighter-sounding.
--scott
--
"C'est un Nagra. C'est suisse, et tres, tres precis."

[Poutnik]

Dne 26.8.2014 v 17:07 Scott Dorsey napsal(a):

> Frank Colessi <n...@nowhere.net> wrote:
>> But the resulting standing wave pattern on a string of variable thickness
>> will likely be similar to a sine wave with increasing amplitude and
>> decreasing frequency along the length of the string (from thin to thick).

Rather variable amplitudes and wavelengths
with different wave phase speed along the string.

As there is requirement of wave continuity,
that cannot be achieved with spatially variable frequency.

>>
>> My guess is that the sound from such standing waves would be harsh due
>> to the closeness of the frequencies generated. I am not sure how the
>> bridge and sounding board would interact with such a standing wave.
>>
>> Such a string could be easily made by wrapping one string with another
>> while increasing the winding distance along the length.
>
> Which is in fact done with some commercial bass strings, and they are in
> fact brighter-sounding.
> --scott
>

--

Poutnik

[JohnF]

In sci.physics Frank Colessi <n...@nowhere.net> wrote:
> On Fri, 25 Jul 2014 21:18:23 +0100, Dave wrote:
>
>> I was reading this:
>> http://www.dailymail.co.uk/sciencetech/article-2703107/Now-THATS-string-theory-Physicist-reveals-science-make-better-guitar-player.html
>> and was wondering about guitar strings with variable thickness along its
>> length.
>

> U_0 is the mass density of the string, which determines the wave velocity:
> v = sqrt(T/u_0).
>
> For a string with variable thickness, u_0 would no longer be a constant
> but would be a function of the position, u_0(x), on the string. The
> resulting wave equation would be non-linear and depending on the form
> of u_0(x) may be impossible to solve explicitly.
>
> But the resulting standing wave pattern on a string of variable thickness
> will likely be similar to a sine wave with increasing amplitude and
> decreasing frequency along the length of the string (from thin to thick).

So how would you solve the modified one-dimensional wave equation?
Typically, with constant linear density, which I'll call lambda,
del^2 y / del x^2 - 1/c^2 del^2 y / del t^2 = 0
where c = sqrt(T/lambda) and T = string tension.
T would still have to remain constant along the string,
and we're still assuming small displacements.
But now lambda = lambda(x), and ditto c.

What are the solutions for some simple lambda(x)'s?
In particular, how is the usual travelling wave solution
modified? Any kind of perturbative solution where
lambda(x) = lambda_0(constant) + small_term(x)?
What else is interesting to say (or ask)?
--
John Forkosh ( mailto: j...@f.com where j=john and f=forkosh )

[sicur...@gmail.com]

This is being experimented with in the piano!!

https://www.youtube.com/watch?v=xcSod-sj2CE&feature=youtu.be

Some really cool stuff. I'm extremely intrigued at what it would sound like on a guitar.